Some Counterexamples to the Central Limit Theorem for Random Rotations

Abstract Fix an irrational number $$\alpha $$ α , and consider a random walk on the circle in which at each step one moves to $$x+\alpha x + or $$x-\alpha - with probabilities 1/2, 1/2 provided current position is x . If observable given we can study process called additive functional of this walk. One formulate certain relations between regularity Diophantine properties implying central limit theorem. It proven here that for every Liouville angle there exists smooth such theorem fails. We construct also fails some analytic observable. For angles counterexample as well. An interesting question remained open.